Weird shapes and lines on solid shapes and low light on cycles
From your picture, the shape glitching looks like Z-fighting. This happens when you have two objects (or faces of the same object) that occupy the same space. If it is Z-fighting, it will probably flicker when you move around in the viewport.

To fix it, try moving the object around. You can also try going into Edit mode, selecting all the vertices, and pressing W > Remove Doubles. (This is also good to do just on general principle—even just canceling a Shift+D will create duplicate vertices in the same spot.)

I don't know about the lighting issue; can you post a picture of the rendered view so I can see what it looks like?